We calculate the exact amenability constant of the centre of ${{\ell }^{1}}\left( G \right)$ when $G$ is a finite group and is either dihedral, extraspecial, or Frobenius with abelian complement and kernel. This is done using a formula that applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk.

We give a constructive proof of the carpenter's theorem due to Kadison. Unlike the original proof, our approach also yields the real case of this theorem.

Let $\left( R,\,\mathfrak{m} \right)$ be a local ring and $\mathfrak{a}$ be an ideal of $R$. The inequalities

$$\text{ht}\left( \mathfrak{a} \right)\,\le \,\text{cd}\left( \mathfrak{a},\,R \right)\,\le \,\text{ara}\left( \mathfrak{a} \right)\,\le \,l\left( \mathfrak{a} \right)\,\le \,\mu \left( \mathfrak{a} \right)$$

are known. It is an interesting and long-standing problem to determine the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.

We prove the following theorem. Suppose that $F\,=\,\left( {{f}_{1}},\,{{f}_{2}} \right)$ is a 2-dimensional, vector-valued modular form on $\text{S}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ whose component functions ${{f}_{1}}$, ${{f}_{2}}$ have rational Fourier coefficients with bounded denominators. Then ${{f}_{1}}$ and ${{f}_{2}}$ are classical modular forms on a congruence subgroup of the modular group.

Let $a$, $b$, and $c$ be primitive Pythagorean numbers such that ${{a}^{2}}\,+\,{{b}^{2}}\,=\,{{c}^{2}}$ with $b$ even. In this paper, we show that if ${{b}_{0}}\,\equiv \,\in \,\,\,\left( \bmod \,a \right)$ with $\text{ }\!\!\varepsilon\!\!\text{ }\,\in \,\left\{ \pm 1 \right\}$ for certain positive divisors ${{b}_{0}}$ of $b$, then the Diophantine equation ${{a}^{x}}\,+\,{{b}^{y}}\,=\,{{c}^{z}}$ has only the positive solution $\left( x,\,y,\,z \right)\,=\,\left( 2,\,2,\,2 \right)$.

We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group, and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $\text{A}\,{{\star }_{\alpha }}\,G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e{{\delta }_{0}}\left( \text{A}\,{{\star }_{\alpha }}\,G \right)e{{\delta }_{0}}$ is a field for all $e\,\in \,E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

The index of a graph $G$ is the maximum eigenvalue of its adjacency matrix $A\left( G \right)$. In this paper we characterize the extremal tree with given domination number that attains the maximum index.

A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Klein bottle if the inclusion $W\,\to \,M$ factors homotopically through a map to $K$. The image of ${{\pi }_{1}}\left( W \right)$ (for any base point) is a subgroup of ${{\pi }_{1}}\left( M \right)$ that is isomorphic to a subgroup of a quotient group of ${{\pi }_{1}}\left( K \right)$. Subsets of $M$ with this latter property are called ${{\mathcal{G}}_{K}}$-contractible. We obtain a list of the closed 3-manifolds that can be covered by two open ${{\mathcal{G}}_{K}}$-contractible subsets. This is applied to obtain a list of the possible closed prime 3-manifolds that can be covered by two open $K$-contractible subsets.

In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if ${{\mathcal{L}}^{\infty }}\left( \mathbb{G} \right)$ contains non-zero compact operators on ${{\mathcal{L}}^{2}}\left( \mathbb{G} \right)$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where ${{\mathcal{L}}^{1}}\left( \mathbb{G} \right)$ has the Radon-Nikodym property.

For relatively prime positive integers ${{u}_{0}}$ and $r$, we consider the least common multiple ${{L}_{n}}\,:=\,\text{lcm}\left( {{u}_{0}},\,{{u}_{1}},\,.\,.\,.\,,\,{{u}_{n}} \right)$ of the finite arithmetic progression $\left\{ {{u}_{k}}\,:=\,{{u}_{0}}\,+\,kr \right\}_{k=0}^{n}$. We derive new lower bounds on ${{L}_{n}}$ that improve upon those obtained previously when either ${{u}_{0}}$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n\,+\,1$ for ${{u}_{0}}$ properly chosen, and is also nearly sharp as $n\,\to \,\infty$.

In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety $X$ over $\mathbf{k}\,=\,\mathbb{C}$. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field $\mathbf{k}$. Secondly we give an isomorphism between the group of Cartier $b$-divisors on the birational class of $X$ and the Grothendieck group of the semigroup of subspaces of rational functions on $X$. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier $b$-divisors and their intersection theory.

In this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of $\Gamma \left( {R}/{I}\; \right)$ and ${{\Gamma }_{1}}\left( R \right)$, and the domination numbers of $\Gamma \left( R \right)$ and $\Gamma \left( R\left[ x,\,\alpha ,\,\delta \right] \right)$, where $R\left[ x,\,\alpha ,\,\delta \right]$ is the Ore extension of $R$, are studied.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ${{\text{ }\!\!\omega\!\!\text{ }}_{1}}$ is hereditarily paracompact.

We give a short proof of the Brascamp–Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prèkopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ${{\mathbb{R}}^{n}}$.

We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

Consider quasianalytic local rings of germs of smooth functions closed under composition, implicit equation, and monomial division. We show that if the Weierstrass Preparation Theoremholds in such a ring, then all elements of it are germs of analytic functions.

An inverse semigroup $S$ is combinatorially factorizable if $S\,=\,TG$ where $T$ is a combinatorial (i.e., $\mathcal{H}$ is the equality relation) inverse subsemigroup of $S$ and $G$ is a subgroup of $S$. This concept was introduced and studied by Mills, especially in the case when $S$ is cryptic (i.e., $\mathcal{H}$ is a congruence on $S$). Her approach is mainly analytical considering subsemigroups of a cryptic inverse semigroup.

We start with a combinatorial inverse monoid and a factorizable Clifford monoid and from an action of the former on the latter construct the semigroups in the title. As a special case, we consider semigroups that are direct products of a combinatorial inverse monoid and a group.